# What is the use of Categorical quantum mechanics?

I recently noticed that Oxford's computer science department has started offering a grad course on Categorical quantum mechanics. Apparently they say that it is relevant for the study of quantum foundations and quantum information, and that it uses paradigms from category theory.

Questions:

1. How exactly does it help in the study of quantum information?

2. Has this formulation actually produced any new results or predictions apart from what our general formulation of quantum mechanics has already done? If so, what are those?

• I think this is primarily opinion based. Also, I don't see any connection to quantum computing. – Norbert Schuch May 4 '18 at 14:51
• @NorbertSchuch Questions about quantum information theory are completely on-topic on this site. And no, such questions are not generally considered "primarily opinion-based" on this site (it's a bit different from the other SE sites). If you feel that it is off-topic consider raising a question on meta. But keep in mind that we have a rather broad scope here. – Sanchayan Dutta May 4 '18 at 14:51
• @NorbertSchuch Generally, if someone posts a question to this site, unless there's good reason to say otherwise, we tend to presume there is a QC connection. As this is a question about a QM course taught by a CS department, I would tend to assume quite a strong connection and would be curious to find out how it's not. In addition, as already mentioned, if you feel something is opinion based, it might be a good idea to bring it up in Quantum Computing Chat, Quantum Computing Meta or flagging, explaining why it's opinion based, in order to improve the question – Mithrandir24601 May 4 '18 at 17:00

This answer is the opinion of someone who is essentially an outsider to "CQM" (= Categorical Quantum Mechanics), but a broadly sympathetic outsider. It should be interpreted as such.

### The motivations of CQM

The motivations of Categorical quantum mechanics are not computation as such, but logic; and not quantum dynamics as such, but foundations of physics. The symptoms of this can be seen in what it describes as its achievements and points of reference, for instance:

• Its results about "completeness" should be interpreted in the same sense as it would in Gödel's Completeness Theorem [sic]: that a set of axioms can perfectly capture a model, which in this case is the model of transformations on a set of qubits expressed in terms of transformations of degrees of freedom expressed in terms of the Z and X eigenbases.

• Occasional comparisons to things like "Rel" (that is: the category of relations, which from a computational point of view is more closely allied to non-deterministic Turing machines than quantum computers) illustrate the fact that they are aware of quantum information theory as being part of a larger landscape of computational theories, where the distinctions between these theories may lead to a robust top-down intuition about what distinguishes quantum theory from other possible dynamical theories of information.

Thus CQM is very much more in a tradition of foundations of physics and the Theory B branch of computer science. So if it does not seem to have developed a lot of "applications" as such, you should not be surprised, because the development of applications is not its primary motivation. (And of course, so far only a very small subset of people in the field are ever really exposed to it.)

### Why CQM might seem a bit obscure

The difference in motivation of CQM to the rest of the field, also reveals itself in the approach which is taken to analysis, in which linear algebra over $\mathbb C$ takes very much a background role.

Linear algebra over $\mathbb C$ is certainly still present in the background, essentially as the target model for CQM. But the usual approach to quantum mechanics in terms of linear algebra over $\mathbb C$ it is seen as potentially obscuring "what is actually going on". And to give the proponents of CQM their due, they have a good argument here: the usual presentation of quantum information theory, starting from vectors over $\mathbb C$ and unitary transformations, working through density operators and CPTP maps, requires a non-trivial amount of work to develop an intuition of what it is for and in what ways it differs (and in what ways it does not differ) from probability theory. It is certainly possible to get that intuition by the usual complex-linear-algebraic approach, but the proponents of CQM would claim that the usual approach is not likely to be the most effective approach.

CQM attempts to put the intuitive meaning front-and-centre, in a mathematically rigorous way. This obligates them to talk about such apparently obscure things as "dagger commutative Frobenius algebras". Of course, such terminology means little to nothing to almost anyone else in the field — but then this is not much different from how quantum information theorists come across to other computer scientists.

This is just the starting point of the potential confusion for an outsider — as those pursuing CQM are in effect mathematicians/logicians with top-down motivations, there is not one single thread of research in CQM, and there is not a sharp boundary between work on CQM and work in higher category theory. This is analogous to the lack of sharp boundary between computational complexity expressed in terms of quantum circuits, quantum communication complexity, query complexity, and the classical version of these topics, along with Fourier analysis and other relevant mathematical tools. Without a clear frame of reference, it can sometimes be a bit confusing as to where CQM begins and ends, but it has in principle as well-defined a notion of scope as any other topic in quantum information theory.

If you wonder why people might like to investigate CQM rather than a more mainstream question in quantum information theory, we should first acknowledge that there are other lines of research in quantum information theory which are not exactly directed towards meaningful impact on anyone else. If we are happy for people to do research into such things as approaches to quantum computation involving physical phenomena which no-one has yet exhibited in the lab [arXiv:1701.05052] or approaches to error correction on closed d-dimensional manifolds for d>2 [arXiv:1503.02065], we should be equally happy to admit other lines of investigation which is somewhat divorced from the mainstream. The justification in each case is the same: that while the arc of theory is long, it bends towards application, and things which are investigated for purely theoretical reasons have a way of yielding practical fruits.

### The use of CQM

On that note: one view of the purpose of paying attention to foundations is to get the sort of insight necessary to solve problems more easily. Does CQM provide that insight?

I think that it is only very recently that the proponents of CQM have seriously considered the question of whether the insights it provides, allow one to obtain new results in subjects which are more in the mainstream of quantum information theory. This is again because the main motivation are the foundations, but recent work has started to develop on the theme of payoffs in the wider field.

There are at least two results which I can point to, which represent ways in which the CQM community has developed results which I would judge to be broadly relevant to the interests of the quantum information community, and in which the results are entirely new:

• Novel techniques for constructing unitary error bases and Hadamard matrices (e.g. [arXiv:1504.02715, arXiv:1609.07775]. These appeared to be of enough interest to the quantum information community that these results were presented as talks in QIP 2016 and 2017 respectively.
• A well-thought out and clear definition of a quantum graph, which recovers the definition of a noncommutative graph from [arXiv:1002.2514] in such a way that makes the relationship to 'classical' graphs clear, allows them to connect to higher algebra, and obtain (Corollary 5.6) a result on the asymptotic density of pairs of graphs for which there is a quantum advantage in pseudo-telepathy games.

As one should expect of abstract mathematical techniques with foundational motivations, there are also payoffs for areas of computer science which are adjacent to quantum information theory:

• Some recent techniques for solving problems in counting complexity regarding the Holant, which are inspired by quantum computation [arXiv:1702.00767], are more specifically inspired by a particular line of investigation into CQM which involved the distinction between GHZ states and W states.

Finally, something which is not yet a result, but which seems a promising direction of research and which in principle does not require category theory to pursue:

• One of the main products of CQM is the ZX-calculus, which one might describe as a tensor-notation which is similar to circuit notation, but which also comes equipped with a formal system for transforming equivalent diagrams to one another. There is a line of investigation into using this as a practical tool for circuit simplification, and for realising unitary circuits in particular architectures. This is based in part on the fact that ZX diagrams are a notation which allows you to reason about tensors beyond just unitary circuits, and which is therefore more flexible in principle.

### Should everyone start using CQM immediately?

Probably not.

As with many things which have been devised for heterodox academic reasons, it is not necessarily the best tool for every question which one might want to ask. If you want to run numerical simulations, chances are you use C or Python as your programming language rather than SML. However, on that same note, just as programming languages developed in earnest by major software firms may in time be informed by ideas which were first developed in such a heterodox academic context, so too might some of the ideas and priorities of CQM eventually filter out to the broader community, making it less an isolated line of investigation than it may seem today.

There are also subjects for which CQM does not (yet) seem to provide a useful way of approaching, such as distance measures between different states or operations. But every mathematical tool has it's limits: I expect that I won't be using quantum channel theory any time soon to consider how to simplify unitary circuits.

There will be problems for which CQM sheds some insight, and may provide a convenient means for analysis. A few examples of such topics are provided above, and it is reasonable to suppose that more areas of application will become evident with time. For those topics where CQM is useful, one can choose whether to take the time to learn how to use the useful tool; apart from that, it's up to you whether or not you are curious enough. In this respect, it is like every other potential mathematical technique in quantum information theory.

### Summary

• If there don't seem to be many novel applications of CQM yet, it's because there aren't — because this isn't the main motivation of CQM, nor have many people studied it.
• Its main motivations are along the lines of foundations of computer science and of physics.
• Applications of the tools of CQM to mainstream quantum information theory do exist, and you can expect to see more as time goes on.